Now the proof is going as follows. Let \(S\)  be a Coble surface (or an Enriques surface) with finite automorphism group.
Then any genus one fibration is extremal, and hence \(S\)  contains an irreducible
effective root.
We take any special genus one fibration on a Coble surface \(S\) .
Together with a special bi-section, we can find a new special genus one fibration which
should be extremal, and hence obtain new effective irreducible roots.
We continue this process to attain a dual graph of effective irreducible roots on \(S\) 
whose reflection group
is of finite index in \({\rm O}({\rm CM}(S))\)  (or \({\rm O}({\rm Num}(S))\)  for Enriques surfaces) or the non-existence of such a surface.
To check the last condition of the finiteness of the index
we use a fundamental result (Theorem REF ) due to Vinberg [1]}. Finally we obtain the dual graphs of
type \({\text{I}}, {\text{II}}, \ldots , {\rm VII}\)  in case of Enriques surfaces and those of type
\({\text{I}}, {\text{II}}, \ldots , {\rm VII, MI, MII}\)  in
case of Coble surfaces (Theorem REF ).
We remark that, as pointed out by Martin [2]},
there are no special elliptic fibrations on an Enriques surface with singular fibers of type
\({\text{I}}_6\) , \(2{\text{I}}_3\) , \(2{\text{I}}_2\)  and the one with four singular fibers of type
\(2{\text{I}}_3, 2{\text{I}}_3, {\text{I}}_3, {\text{I}}_3\) , where \(2{\text{I}}_3\)  means that the fiber of type \({\text{I}}_3\)  is multiple.
On the other hand the author [3]}
used some property of automorphisms of complex \(K3\)  surfaces to exclude these fibrations.
Allowing these two fibrations we obtain the dual graphs of type
MI and MII.
In this process we use the following Martin's idea using the theory of Mordell-Weil lattice (e.g. Schütt and Shioda [4]}).
For a special genus one fibration \(p:S\rightarrow {\bf P}^1\)  with a \((-2)\) -curve as its bi-section,
one can construct the Jacobian fibration \(j(p)\)  of \(p\)  concretely.
By applying the theory of Mordell-Weil lattices
one can determine the sections of \(j(p)\)  and their intersections with reducible fibers.
Thus we can find a new irreducible effective root on \(S\)  which is helpful to determine the dual graph
of effective irreducible roots.
